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Adsorption Dynamics. Extension of Self-Consistent Field Theory to Dynamical Problems Ryuichi Hasegawa*,†,‡ and Masao Doi‡ Yokkaichi Research Center, Mitsubishi Chemical Company, Toho-cho, Yokkaichi, Mie 510, Japan, and Department of Applied Physics, Nagoya University, Chikusa-ku, Nagoya, Aichi 464, Japan Received November 1, 1996; Revised Manuscript Received March 10, 1997X ABSTRACT: We propose a new simulation scheme to describe polymer dynamics at interfaces. Using this scheme, we studied adsorption dynamics in a polymer solution. We calculated the evolution of the density profiles and the polymer conformation. The numerical results are qualitatively consistent with recent experimental observations.
Introduction The process whereby polymers in solution adsorb on a bare surface typically involves adsorption-desorption dynamics and has attracted considerable attention both theoretically and experimentally.1-6 It is known that the process involves a large structural charge of the adsorbed layer: the conformation of the adsorbed polymers change from a random-coil to a stretched structure as the adsorption proceeds. Previous theories dealt with the dynamics in the limiting cases before and after the adsorbed layer was formed but did not examine the structural change itself. In this paper, we study this problem using a new simulation scheme. The method is a dynamical version of the mean field theory which was first proposed by Fraaije7 for the microphase separation kinetics of block copolymers. The scheme calculates the free energy density at each time step in solving the diffusion equation. For the present problem, the results provide detailed information on the evolution of the structure of the adsorbed layer including the segment density and the fractions of trains and loops, etc. Formulation We consider a bare surface with no adsorbed polymer which is inserted into a polymer solution at time t ) 0 and ask how the density profile of polymer segments changes in time as the polymers adsorb on the surface. We assume the z coordinate is perpendicular to the surface with its origin located at the surface. Let Φp(z,t) and Φs(z,t) be the dimensionless densities of polymer segment and solvent molecules respectively at time t and position z. The polymer segment density Φp(z,t) consists of two parts, Φa(z,t), the segment density of adsorbed polymers which have at least one segment adsorbed on the surface, and Φf(z,t), the segment density of free polymers which have no adsorbed segments:
Φp(z,t) ) Φa(z,t) + Φf(z,t)
(1)
The time evolution of Φa(z,t) and Φf(z,t) is determined by the diffusion of polymer segments driven by the gradient of the chemical potential and the adsorptiondesorption dynamics taking place at the surface. Accounting for these two processes in the general case is * To whom correspondence should be addressed. Phone: 81593-45-7096. FAX: 81-593-45-7044. † Mitsubishi Chemical Co. ‡ Nagoya University. X Abstract published in Advance ACS Abstracts, May 1, 1997.
S0024-9297(96)01618-X CCC: $14.00
a complex problem. In this paper, we consider the special case that the adsorption-desorption dynamics takes place much faster than the diffusion dynamics. In such a case, the exchange between the adsorbed polymer and the free polymer takes place very frequently, and they are in equilibrium. Therefore in determining the time evolution of the segment density, we need not distinguish segments of the adsorbed polymer and those of the free polymer: we can first determine the time evolution of the total segment density Φp(z,t), and then calculate Φa(z,t) and Φf(z,t) from the condition that they are in equilibrium for a given total segment density profile Φp(z,t). We assume that Φp(z,t) and Φs(z,t) satisfies the incompressibility condition.
Φp(z,t) + Φs(z,t) ) 1
(2)
Therefore, the concentration profile is described by a single field Φ(z,t).
Φ(z,t) ) Φp(z,t) - Φs(z,t)
(3)
Let F[Φ] be the free energy of the whole system for a given density profile Φ(z). The diffusion of polymer segments (and solvent molecules) is driven by the gradient of the chemical potential δF/δΦ. Thus we may assume that the time evolution of Φ(z,t) is determined by the following diffusion equation
∂F ∂Φ ) ∇M∇ ∂t δΦ
(4)
where M is the mobility coefficient, which is in general a function of the local segment density. For the sake of simplicity, we use a “free draining” picture in this paper and assume that M is constant. The free energy functional F[Φ] is written as
∫0∞χΦpΦs dz - χsΦp(0) ∞χ ) F0[Φ] + ∫0 (1 + Φ)(1 - Φ) dz 4
F[Φ] ) F0[Φ] +
χs (1 + Φ(0)) 2
(5)
where χ (and χs) are dimensionless parameters, representing the interaction between polymer segments and solvent molecules (and between polymer segments and the surface), and F0[Φ] is the conformational entropy © 1997 American Chemical Society
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part of the free energy: F0[Φ] is the free energy of an athermal system in which χ and χs are equal to zero. To calculate F0[Φ], we consider an athermal system which has a density profile Φ(z,t) which is in equilibrium subject to a hypothetical external potential field V(z,t). If the external potential field V(z,t) is known, the functional derivative δF0/δΦ is given by
δF0 ) -V δΦ
(6)
Thus from eqs 5 and 6, δF/δΦ is calculated as
χs χ δF ) -V - Φ - δz0 δΦ 2 2
(7)
To obtain an expression for the potential field V(z,t), we assume that the polymer segments and the solvent molecules are in equilibrium under hypothetical external potential field Vp(z,t) and Vs(z,t) each acting on the polymer segments and the solvent molecules respectively.
V(z,t) ) Vp(z,t) - Vs(z,t)
Figure 1. Time development of the amount of adsorbed polymer when a bare surface is inserted into a polymer solution of concentration of Φb, where the chain length N ) 30, polymer-solvent interaction parameter χ ) 0, 0.5, and polymer-surface interaction parameter χs ) 1.0.
(8)
We now find the potentials Vp(z,t) and Vs(z,t) which give the density profiles Φp(z,t) and Φs(z,t) for an equilibrium athermal system. It is easy to obtain Vs(z,t). Under an external potential Vs(z,t), the equilibrium density profile of the solvent is given by
Φs(z,t) ) (1 - Φb) exp(-Vs(z,t))
(9)
where Φb is the bulk concentration of the polymer Φb ) limzf∞ Φp(z,t). Thus the potential field Vs(z,t) is given by
(
Vs(z,t) ) -ln
Φs(z,t)
(1 - Φb)
)
(10)
To obtain Vp(z,t) we use the self-consistent field theory for polymer statistics. The polymer segment density Φp(z,t) satisfies the following equation
Φp(z,t) )
Φb N
∫0NΨp(z,n,t)Ψp(z,N - n,t) dn
(11)
where Ψp(z,n,t) is the solution of the following Schro¨dinger-type equation
∂Ψp a2 2 ) ∇ Ψp - VpΨp ∂n 6
(12)
where a is the segment size with the “initial” condition and the boundary condition respectively
Ψp(z,0,t) ) 1
(13)
Ψp(z,n,t) ) 0 z < 0
(14)
From the above set of equations eqs 11-14, the potential field Vp(z,t) can be calculated using the discrete version of the self-consistent field theory as described in ref 9. Equations 4-7 determine the time evolution of Φ(z,t). The initial condition is that the segment density is uniform at time t ) 0, i.e., Φp(z,0) ) Φb or Φ(z,0) ) 2Φb - 1. The boundary condition is that there is no flux at
Figure 2. Time evolution of the thickness of adsorbed layer, ξ(t). The solid line represents the case of χ ) 0.5 and the dashed line represents the case of χ ) 0.0.
the wall, i.e. ∂(δF/δΦ)/∂z ) 0 at z ) 0. The unit of length is taken to be the segment size a, and the unit of time, the segmental diffusion time a2/M, which approximately corresponds to the time it takes for a segment to diffuse a distance a by cooperative diffusion. Solution of the time evolution equation gives Φ(z,t), Φp(z,t), and Vp(z,t). If these quantities are known, the details of the conformational statistics can be easily calculated. For example, the density profile Φf(z,t) of the free polymer is calculated from the following set of equations:
Φf(z,t) )
Φb N
∫0NΨf(z,n)Ψf(z,N - n) dn
(15)
∂Ψf a2 2 - ∇ Ψf - VpΨf ∂n 6
(16)
Ψf(z,0) ) 1 - δz0
(17)
Ψf(0,n) ) 0
(18)
The density profile Φa(z,t) of the adsorbed polymer is calculated using
Φa(z,t) ) Φp(z,t) - Φf(z,t)
(19)
Similarly, other statistical details, such as the average length and numbers of trains, loops, and tails etc, can be calculated using the method of Scheutjen and Fleer.10 Results and Discussion As illustration of the theory, we present results for χ ) 0 (good solvent), and χ ) 0.5 (Θ solvent). The values
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Figure 3. Time evolution of the density profile during adsorption. From top to bottom, are shown the density profiles of (a) the segments of adsorbed polymer, (b) the segments of free polymer, and (c) solvent molecules. The constants are χ ) 0.5, N ) 30, Φb ) 0.1, and χs ) 1.0.
of the other constants are N ) 30, Φb ) 0.1 and χs ) 1.0. Figure 1 shows the time evolution of the adsorbed amount
Γ)
∫0∞Φa(z) dz
(20)
Initially, adsorption takes place very rapidly, but then it slows down considerably. This behavior is similar to that obtained experimentally.5,6 Clearly, the equilibrium value of the adsorbed amount is larger in a poor solvent than in a good solvent. Figure 2 shows the time evolution of the thickness of the adsorbed layer
∫0∞zΦa(z,t) dz ξ(t) ) ∞ ∫0 Φa(z,t) dz
(21)
It is seen that ξ(t) first decreases and then increases again: The reason for this behavior will be discussed below. Figure 3 shows the time evolution of the density profiles of each component, Φa(z,t) (the adsorbed polymer), Φf(z,t) (the free polymer), and Φs(z,t) (the solvent molecules). The equilibrium density profile in Figure 3 is calculated using the usual static self-consistent field theory. It is seen that the profile of the adsorbed layer approaches the equilibrium shape quickly, but the density profile of the free polymer varies very slowly in time. Also notice that in the intermediate stage (t = 10), there appears a peak in the solvent density at the edge of the adsorbed layers z = 4. The peak in the solvent density corresponds to the depletion of the polymer segment at the edge of the adsorbed layer.
Figure 4. Time evolution of the average quantities of tails, trains, and loops. From top to bottom are shown (a) the average length, (b) the average number per chain, and (c) the fraction of segments belonging to tails, trains, and loops. The solid line represents the case of χ ) 0.5, and the dashed line represents the case of χ ) 0.0.
Figure 4 shows some structural changes which take place in the adsorbed polymer. Following ref 11, we call a consecutive sequence of adsorbed segments a train, the part between the successive trains, a loop, and the part between a train and the end of the polymer, a tail. Figure 4a shows the time evolution of the average length (the number of segments) of trains, loops, and tails of the adsorbed polymers. Notice that the length of the trains has a maximum at t = 10, while the length of the tails and loops has minimum at about the same time. Similar behavior is observed in the fraction of trains, tails and loops (Figure 4c). The behavior shown in Figures 2-4 suggests the following picture of the adsorption dynamics. The adsorption process consists of two stages. In the first stage (0 < t j 10), the polymers which happen to have segments near the surface are adsorbed on the bare surface. If this happens, the other segments connected to those segments are also adsorbed on the surface. This results in an increase in the average length of the trains and a decrease in the length of tails and loops. During this process, the adsorbed polymers become more flattened, and the thickness of the adsorbed layer decreases. Consequently, there appears a polymer depletion layer near the surface which corresponds to the peak in the solvent density in Figure 3c. In the second stage of adsorption (t J 10), the free polymers penetrate into the adsorbed layer. The second stage is very slow compared with the first stage, since the free polymers have to diffuse into the adsorbed layer and reach the surface. As this happens, some adsorbed segments in the existing layer are replaced by the segments of free polymer. Accordingly, the average length of trains decreases, and the average lengths of
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tails and loops increase. The picture described here is qualitatively consistent with the recent experimental observation by Granick et al.12 They observed that as adsorption proceeds, the number of segments which are oriented perpendicularly to the wall first decreases and then increases again. If we assume that segments in trains are parallel to the wall and segments in loops and tails are (in the first approximation) perpendicular to the wall, their results are consistent with our results in Figure 4c. The present formulation is useful for other problems such as the replacement dynamics5,6 or the chemical reaction between polymers13 where the dynamics is coupled with the conformational changes of the polymers. Such problems are now being studied, and the results will be published in the future.
References and Notes
Acknowledgment. The authors thank Prof. T. Kawakatsu for helpful discussions and Prof. S. Granick for useful comments.
(13)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
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